The expansion problem of anti-symmetric matrix under a linear constraint and the optimal approximation

[摘要] This paper mainly discusses the following two problems: Problem I. Given A is an element of R-nxm, B is an element of R-mxm, X-0 is an element of ASR(qxq) (the set of q x q anti-symmetric matrices), find X is an element of ASR(nxn) such that A(T)XA = B, X-0 = X([1 : q]), where X([1 : q]) is the q x q leading principal submatrix of matrix X. Problem II. Given X* is an element of R-nxn, find (X) over cap is an element of S-E such that parallel to X* - (X) over cap parallel to = min(X is an element of SE) parallel to X* - X parallel to, where parallel to center dot parallel to is the Frobenius norm, and SE is the solution set of Problem I. The necessary and sufficient conditions for the existence of and the expressions for the general solutions of Problem I are given. Moreover, the optimal approximation solution, an algorithm and a numerical example of Problem II are provided. (c) 2005 Elsevier B.V. All rights reserved.

[发布日期] 2006-12-01

[关键词] anti-symmetric matrix;linear constraint;Frobenius norm;optimal approximation